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Using time-resolved magneto-optical Kerr effect (TR-MOKE) microscopy, we demonstrate surface-acoustic-wave (SAW) induced resonant amplification of intrinsic spin-wave (SW) modes, as well as generation of new extrinsic or driven modes at the SAW frequency, in a densely packed two-dimensional array of elliptical Co nanomagnets fabricated on a piezoelectric LiNbO 3 substrate. This system can efficiently serve as a magnonic crystal (MC), where the intrinsic shape anisotropy and the strong inter-element magnetostatic interaction trigger the incoherent precession of the nanomagnets' magnetization in the absence of any bias magnetic field, giving rise to the ‘intrinsic’ SW modes. The magnetoelastic coupling leads to a rich variety of SW phenomena when the SAW is launched along the major axis of the nanomagnets, such as 4–7 times amplification of intrinsic modes (at 3, 4, 7 and 10 GHz) when the applied SAW frequencies are resonant with these frequencies, and the generation of new extrinsic modes at non-resonant SAW frequencies. However, when the SAW is launched along the minor axis, a dominant driven mode appears at the applied SAW frequency. This reveals that the magnetoelastic coupling between SW and SAW is anisotropic in nature. Micromagnetic simulation results are in qualitative agreement with the experimental observations and elucidate the underlying dynamics. Our findings lay the groundwork for bias-field free magnonics, where the SW behavior is efficiently tuned by SAWs. It has important applications in the design of energy efficient on-chip microwave devices, SW logic, and extreme sub-wavelength ultra-miniaturized microwave antennas for embedded applications.